I propose a novel framework that integrates stochastic differential equations (SDEs) with deep generative models to improve uncertainty quantification in machine learning applications involving structured and temporal data. This approach, termed Stochastic Latent Differential Inference (SLDI), embeds an Itô SDE in the latent space of a variational autoencoder, allowing for flexible, continuous-time modeling of uncertainty while preserving a principled mathematical foundation. The drift and diffusion terms of the SDE are parameterized by neural networks, enabling data-driven inference and generalizing classical time series models to handle irregular sampling and complex dynamic structure. A central theoretical contribution is the co-parameterization of the adjoint state with a dedicated neural network, forming a coupled forward-backward system that captures not only latent evolution but also gradient dynamics. I introduce a pathwise-regularized adjoint loss and analyze variance-reduced gradient flows through the lens of stochastic calculus, offering new tools for improving training stability in deep latent SDEs. My paper unifies and extends variational inference, continuous-time generative modeling, and control-theoretic optimization, providing a rigorous foundation for future developments in stochastic probabilistic machine learning.

翻译：本文提出了一种融合随机微分方程与深度生成模型的新型框架，以提升涉及结构化时序数据的机器学习应用中的不确定性量化能力。该框架被命名为随机隐变量微分推断，通过在变分自编码器的隐空间中嵌入伊藤随机微分方程，实现了灵活、连续时间的不确定性建模，同时保持了严谨的数学基础。随机微分方程的漂移项与扩散项由神经网络参数化，支持数据驱动的推断过程，并将经典时间序列模型推广至不规则采样与复杂动态结构的场景。核心理论贡献在于通过专用神经网络对伴随状态进行协同参数化，构建了耦合的前向-后向系统，该系统不仅能捕捉隐状态演化，还能刻画梯度动态特性。本文提出了路径正则化的伴随损失函数，并通过随机微积分视角分析了方差缩减的梯度流，为提升深度隐变量随机微分方程的训练稳定性提供了新工具。本工作统一并拓展了变分推断、连续时间生成建模与控制理论优化，为随机概率机器学习的未来发展奠定了严格的理论基础。